The co-degree density of the Fano plane

It is shown that every n vertex triple system with every pair of vertices lying in at least (1/2 + o(1))n triples contains a copy of the Fano plane. The constant 1/2 is sharp. This is the first triple system for which such a (nonzero) constant is determined. Let X be a finite set. An r-graph F with vertex set X is a family of r-element subsets of X. These subsets are called edges, and X is written as V (F ). When there is no confusion, we denote an edge {x, y, z} as xyz. Given a family of r-graphs F , the Turán number ex(n,F) is the maximum number of edges in an n vertex r-graph containing no member of F . The Turán density of F is defined as π(F) := limn→∞ ex(n,F)/ ( n r ) . If F = {F}, we write π(F ) instead of π({F}). For r > 2, computing π(F ) when it is nonzero is notoriously hard, even for very simple r-graphs F (see [3] for a survey of results). Determining the Turán density of complete r-graphs is a fundamental question about set-systems. In fact, this is not known in any nontrivial case when r ≥ 3. In the past few years, there has been some progress on related problems. The Fano plane is the projective plane PG(2, 2) consisting of seven vertices (points) and seven edges (lines). Alternatively, we can write it as {yaa′, ybb′, ycc′, abc′, ab′c, a′bc, a′b′c′}. The starting point of these recent developments was de Caen and Füredi’s breakthrough proof [1] of Sós’ 1973 conjecture that π(F) = 3/4, where F is the Fano plane (see also Füredi-Simonovits [4] and Keevash-Sudakov [5] for exact results and further extensions). The co-degree of vertices x, y in a hypergraph G is the number of edges containing both x and y. Define c(G) to be the minimum co-degree over all pairs of vertices in G. A natural problem is to ask what minimum co-degree guarantees a copy of some forbidden configuration. The first question in this area (to the authors knowledge) was posed by S. Abbasi (personal communication with V. Rödl), who (implicitly) asked whether, for an n vertex 3-graph G, the condition c(G) ≥ n/2 suffices ∗Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, IL 60607. Research supported in part by National Science Foundation grant DMS-0400812 and an Alfred P. Sloan Research Fellowship; email: [email protected] 2000 Mathematics Subject Classification: 05C35, 05C65, 05D05